(e.g., Kilbas, Srivastava and Trujillo [1]}, p.138).
Indeed, we can directly verify that \(u\)  given by (1.10) satisfies (1.9):
\(& D_t^{\alpha }J^{\alpha }f(t)= \frac{1}{\Gamma (\alpha )\Gamma (1-\alpha )}\frac{d}{dt}\int ^t_0 (t-s)^{-\alpha } \left( \int ^s_0 (s-\xi )^{\alpha -1}f(\xi ) d\xi \right) ds\\=& \frac{1}{\Gamma (\alpha )\Gamma (1-\alpha )}\frac{d}{dt}\int ^t_0 f(\xi ) \left( \int ^t_{\xi } (t-s)^{-\alpha }(s-\xi )^{\alpha -1}ds \right) d\xi \\= & \frac{d}{dt}\int ^t_0 f(\xi ) d\xi = f(t) \quad \mbox{for$f\in L^1(0,T)$}.\) 
